Never mind the hyperbolics:
nonparametric analysis of timeinconsistent preferences
IFS Working Paper W14/17
Laura Blow
Martin Browning
Ian Crawford
The Institute for Fiscal Studies (IFS) is an independent research institute whose remit is to carry out
rigorous economic research into public policy and to disseminate the findings of this research. IFS
receives generous support from the Economic and Social Research Council, in particular via the ESRC
Centre for the Microeconomic Analysis of Public Policy (CPP). The content of our working papers is
the work of their authors and does not necessarily represent the views of IFS research staff or
affiliates.
Never Mind the Hyperbolics: Nonparametric Analysis of
Time-Inconsistent Preferences
Laura Blow∗, Martin Browning†& Ian Crawford‡
Abstract
We investigate necessary and sufficient nonparametric conditions for the quasi-hyperbolic
consumer. These turn out to be quite tractable. We investigate the performance of this
model compared to the standard exponential discounting model using consumer panel data.
Key Words: Quasi-hyperbolic discounting, revealed preference.
JEL Classification: C43, D11.
Acknowledgements: We are very grateful for financial support from the European
Research Council grant ERC-2009-StG-240910-ROMETA, the Leverhulme Trust grant
F/08 519/E and the ESRC Centre for Microeconomic Analysis of Public Policy at IFS
(grant RES-544-28-5001). We thank seminar audiences at Oxford, University College
London, Universitat Pompeu Fabra and Bristol for their patience and helpful comments.
1
Introduction
What are the necessary and sufficient nonparametric/revealed preference empirical conditions
for quasi-hyperbolic consumption behaviour? This paper explores this in the elementary,
choice-revealed preference tradition of Samuelson (1948), Houthakker (1950) and Afriat (1967).
We describe these conditions for the leading forms of the model and discuss how to implement
them and also how to make sensible empirical comparisons with alternative models including
the standard exponential discounting model. We also provide an empirical application using
a large, nationally representative panel household consumption survey.
In recent years quasi-hyperbolic discounting has begun to prove itself a major challenge to
the traditional exponential discounting framework. The standard assumption of exponential
discounting with its constant discount rate is, of course, a parsimonious one since it allows
∗
Institute for Fiscal Studies. 7 Ridgmount Street, London, WC1E 7AE. E-mail: laura b@ifs.org.uk.
Nuffield College and University of Oxford, Department of Economics, Manor Road Building, Manor Road,
Oxford OX1 3UQ, United Kingdom. E-mail: martin.browning@economics.ox.ac.uk.
‡
IFS, Nuffield College and University of Oxford, Department of Economics, Manor Road Building, Manor
Road, Oxford OX1 3UQ, United Kingdom. E-mail: ian.crawford@economics.ox.ac.uk.
†
1
a person’s time preference to be summarised as a single parameter and a useful one too
since it makes the strong prediction that the consumer’s intertemporal preferences are timeconsistent. However a certain amount of evidence has accrued which indicates that people often
do not behave in a time-consistent manner.1 Indeed Samuelson (1937), when he introduced
the exponential discounted utility model, in the same breath disavowed it on precisely these
grounds:
“our equations hold only for an individual who ... will still make at each instant
the same decision with respect to expenditure as he would have, if at the beginning
of the period he were to decide on his expenditure for the whole period. But the
fact that this is so is in itself, a presumption that individuals do not behave in
terms of our functions.”
2
Despite Samuelson’s manifest misgivings, the exponential form was too useful to be ignored
and was quickly adopted as the standard model of intertemporal consumer behaviour. This
state of affairs lasted until the late 1990’s when economists began first to question it and then
to abandon it in favour of the quasi-hyperbolic discounting model3,4 . This process is (pungently) described in Rubinstein (2003). The model has now been widely adopted and applied,
it would seem with some success, to describe a range of phenomena from the role of illiquid
assets as commitments5 , the excess sensitivity of consumption to income and the retirement
savings puzzle6 , the simultaneous holding by households of high pre-retirement wealth, low
liquid assets and high credit-card debt7 , labour supply and welfare program participation8 ,
procrastination in a number of contexts9 , addiction/habit formation10 , information acquisition11 and the Phillips curve12 . This is a far from exhaustive list of applications but it serves
to give a sense of the breadth of topics to which the model has been fruitfully applied. The
emerging consensus seems to be that, compared the the exponential model, quasi-hyperbolic
discounting better fits the stylised facts regarding individuals’ intertemporal behaviour and
the experimental evidence. Indeed, Frederick et al (2001, p361) conclude that “the collective
evidence ... seems overwhelmingly to support hyperbolic discounting”.
1
See Frederick et al (2002) for a survey of the empirical literature.
Samuelson (1937, p160)
3
Samuelson foreshadowed precisely this in his original article, writing: “Actually, however, as the individual
moves along in time there is a sort of perspective phenomenon in that his view of the future in relation to his
instantaneous time position remains invariant, rather than his evaluation of any particular year (e.g. 1940).
This relativity effect is expressed in the behaviour of men who make irrevocable trusts, in the taking out of life
insurance as a compulsory savings measure, etc.” (Samuelson, (1937, p. 160)).
4
Strotz (1955-1956) considered non-exponential discounting and Phelps and Pollak (1968), and then Elster
(1979) studied the, now firmly established, (β, δ) form. Laibson (1994,1997,1998) Harris and Laibson (2001) in
particular have analysed the implications of this form extensively.
5
Laibson (1997)
6
Laibson (1998)
7
Angeletos et al. (2001)
8
Fang and Silverman (2009).
9
Fischer (1999) and O’Donoghue and Rabin (1999, 2001)
10
O’Donoghue and Rabin (2000), Gruber and Koszegi (2000), and Carrillo (1999).
11
Carrillo and Mariotti (2000) and Benabou and Tirole (2000).
12
Graham and Snower (2007).
2
2
In this paper we study quasi-hyperbolic consumption behaviour from a revealed preference
perspective. When researchers are armed with a theory about what the individual may be
maximising they typically proceed in one of two ways. The first is the structural econometric
approach in which the implications of the theory are described in terms of the properties
of certain key functions (compensated demand curves, Euler equations and the like). Since
these functions are never observed directly they need to be estimated from the data so the
econometrician appends a statistical structure to the economic model in order to account for
the fact that the economic theory, as expressed through the structural equations, does not
perfectly fit the data. This extra structure generally entails statistical assumptions regarding
the joint distribution of unobserved explanatory variables and unobservable error terms. The
art of structural modelling thus mainly lies in getting this statistical aspect right, because the
source and the properties of these unobserved and unobservable factors can have a critical
impact on the estimation results (see for example, McElroy (1987), Brown and Walker (1989),
Lewbel (2001)).
The alternative is empirical revealed preference analysis. This is the approach which we
adopt. Rather than describing the implications of the theory in terms of estimated structural
equations, empirical revealed preference uses systems of inequalities which depend neither on
the form of structural functions nor on unobservables. Statistical error terms and special
assumptions about the functional form of the economic model may be added but it is not
an essential requirement of the approach. The classic example is the Generalised Axiom of
Revealed Preference (GARP) which is the necessary and sufficient condition for the standard
utility maximisation model with competitive linear pricing. GARP is a simple system of inequalities (or an equivalent linear program) which only involves the observed prices and choices
and which is “nonparametric” in the sense that it assumes nothing about the functional form
of preferences. By Afriat’s Theorem13 if the data satisfy GARP, then they are consistent with
the hypothesis that they were generated by a rational consumer with well-behaved preferences.
If they are not consistent with GARP then the data are not rationalisable by the standard neoclassical utility maximisation model and the model, or perhaps the data, need modification.
Thus GARP exhausts the empirical implications of the standard consumer choice model.
In this paper we are, in essence, trying to see whether there is a GARP-like condition for
the quasi-hyperbolic consumption model. The answer is a non-obvious - or at least was to
us. This is because empirical revealed preference conditions usually exploit some consistency
property in the individual’s behaviour which is implied by the model. The quasi-hyperbolic
model, however, explicitly implies behaviour which is inconsistent in some respects. The
answer is also of interest: if a revealed preference characterisation does not exist then the
model is not falsifiable in the same way that, for example, the exponential discounting model
is (see Browning (1989) for a version of Afriat’s Theorem which applies to the exponential
model); if such a characterisation does exist then this is an important supportive result for
one of the leading models in the Behavioural canon.
13
Afriat (1967) , Diewert (1973) and Varian (1982).
3
We present necessary and sufficient conditions for the quasi-hyperbolic consumption14
model and show that the associated empirical tests are rather straightforward. We also show
a number of interesting related results which describe, in general terms, what can be learned
about the quasi-hyperbolic model from the data without making strong identifying assumptions. We show that whilst it is possible to distinguish the sophisticated quasi-hyperbolic
consumer from his exponential cousin, it is not possible to disentangle the hyperbolic individual’s two discount rates. In contrast, we also show that using observables alone one cannot
distinguish between the naive quasi-hyperbolic consumer and an exponential consumer with
a very high discount rate - the same behaviour is rationalisable by both models. We then
show how our results relate to those of Afriat (1967), Browning (1989) and others. We also
investigate how the quasi-hyperbolic consumption model performs compared to the exponential model using a large, consumption panel data set. In this respect it is worth noting that
it is often claimed that the hyperbolic model outperforms the exponential model empirically
(see many of the papers cited above). We investigate this and it is indeed the case. But
of course it could hardly be otherwise: the quasi-hyperbolic model is more general than the
exponential model because, compared to the exponential model it contains a free parameter.
It therefore necessarily fits the data no worse than the standard model. Merely claiming that
the hyperbolic model is better in terms of fit is not a rational reason to prefer it. In this paper
we consider this problem in some detail and suggest a different way of comparing models using
their empirical informational content.
2
Necessary and Sufficient Conditions for the Quasi-hyperbolic
Consumption Model
2.1
The self-aware individual
We are interested in the empirical implications of the quasi-hyperbolic discounting model for
a finite dataset of interest rates, spot prices and purchases of K goods {rt , pt , ct }t=1,...,T for a
mortal, self-aware individual who has no ability to commit to consumption plans. We begin
by considering the perfect-foresight case and discuss the extension to uncertainty (which is
straightforward) in due course. We consider what we take to be a standard version of the
14
We focus on the consumption model for a number of reasons. The first that consumption behaviour is
intrinsically important in both macro and microeconomics (we take this as read). The second is that we would
like to develop methods which can be applied to readily-available real world data. The strongest evidence
in favour of hyperbolic discounting behaviour often comes from the lab (Thaler (1981), Benzion, Rapoport
and Yagil (1989), Kirby and Herrnstein (1996)) but the methodological status of lab studies in economics
is far from uncontroversial (see for example Rubinstein (2001) and Levitt and List (2007)). We are therefore
interested in investigating hyperbolic discounting using data from standard, widely-available real world datasets
on consumption behaviour of the sort which have been used for many years to estimate Euler equations like the
Consumer’s Expenditure Survey (CEX) in the US or the Family Expenditure Survey (FES) in the UK. Thirdly,
and perhaps most importantly, consumption behaviour is a good focus of study for our purposes because the
data are often particularly abundant. Time-inconsistency may be an important feature of households’ decisions
to invest in a house or an individual’s decision to investment in education, but such decisions are made rather
infrequently. Consumption decisions, on the other hand, are made all the time by households and so provide
an excellent data-rich context for studying intertemporal models.
4
hyperbolic consumption model in which the individual is modelled as a composite of temporal
selves indexed by their respective periods of control, t = 0, 1, 2, ..., T , over a consumption
decision. During their period of control, self t observes all past consumption decisions, and
the current level of financial wealth At and chooses a consumption bundle for period t, such
that
p0t ct + St = Yt + At
Savings/dis-savings are denoted St . Self t + 1 then inherits wealth equal to
At+1 = (1 + rt+1 ) St
The game continues, with self t + 1 in control. We assume that AT = 0. The payoff for the
t’th player of this game is
U (ct , ct+1 , ..., cT ) = u (ct ) + β
T −t
X
δ i u (ct+i )
i=1
where u is a concave, continuous and differentiable instantaneous utility function. Optimising
behaviour in this model can be characterised using the first order condition on the equilibrium
path (see Harris and Laibson,(2001)). The following is a slight generalisation of their result
which allows for multiple consumption goods - which will feature in our application.
Lemma 1 (Harris and Laibson (2001)). On the equilibrium path:
"
#−1
t
K k
Y
X
∂u
∂c
= δ −t λρkt
1 − (1 − β)
pki i
∂Ai
∂ckt
i=1
k=1
∀k
and the corresponding Euler equation is
!#
"
K
X
∂ckt+1
∂u
pkt
∂u
k
= δ (1 + rt+1 ) k
1 − (1 − β)
pt+1
k
∂At+1
∂ct
pt+1
∂ckt+1
k=1
where ρkt = pkt /
Qt
i=1 (1
∀k
+ ri ) denotes the discounted price of the kth good, λ is a strictly
positive constant (the marginal utility of wealth in period 0), β and δ lie in the interval [0, 1]
and ∂ckt /∂At is the marginal propensity to consume out of current wealth for the kth good.
Proof. See the Appendix for this and all subsequent proofs.
Note that this first order condition and Euler equation simplify to the standard exponential
discounting case when β = 1. Both expressions are slight generalisations of those in Harris
and Laibson (2001)15 and readily reduce to those in Harris and Laibson (2001) if we consider
15
Note that in deriving the first order condition and the Euler equation we follow the heuristic method
of Harris and Laibson (2001) - that is we simply make the necessary assumptions regarding the smoothness/differentiability of demands. As Harris and Laibson (2001) show, the same results can be derived under
less benign conditions. Readers are referred to their paper for this derivation.
5
a single consumption good, no inflation and a fixed interest rate. The object
K
X
pkt+1
k=1
∂ckt+1
∂At+1
!
is the period t + 1 marginal propensity to spend out of wealth and, assuming that demands
are normal, it lies between zero and one. Thus we can rewrite the hyperbolic Euler equation
as
∂u
pkt ∂u
∗
=
δ
(1
+
r
)
t+1
∂ckt
pkt+1 ∂ckt+1
where
"
∗
δ =δ 1−
K
X
k=1
pkt+1
∂ckt+1
∂At+1
!#
"
+ δβ
K
X
k=1
pkt+1
∂ckt+1
∂At+1
!#
is sometimes termed the “effective discount factor”. This is a weighted average of the shortrun discount factor δβ, and the long-run discount factor δ. This captures the conflict between
the short-term and the long-term self because what the future self does matters to the current
self. The larger the future self’s marginal propensity to spend the more important the shortrun discount factor and the current marginal utility is lower relative to the future than it
would otherwise be in the standard case. This is due to the tendency of the future self to
over-consume, as far as the current self is concerned, and thus to reduce that period’s marginal
utility of consumption relative to subsequent periods.
Following Browning (1989) we define what it means for the model to rationalise the data
as follows.
Definition 1 The self-aware quasi-hyperbolic discounting model rationalises the data {rt , pt , ct }t∈τ
if there exists a locally non-satiated, differentiable and concave instantaneous utility (felicity)
function u (.) and constants λ > 0 , β ∈ [0, 1], δ ∈ [0, 1] and {µt ∈ [0, 1]}t∈τ such that
t
∂u
1 kY
1
= λ t ρt
k
δ
[1 − (1 − β) µi ]
∂ct
i=1
where ρkt = pkt /
∀k
PK k ∂cki (1
+
r
)
and
µ
=
i
i
i=1
k=1 pi ∂Ai denotes the marginal propensity to spend
Qt
in period i.
This says that the data are consistent with the theory if there exists a well-behaved instantaneous utility/felicity function, the derivatives of which satisfy the hyperbolic first order
conditions (or equivalently the hyperbolic Euler equation) on the equilibrium path. If such a
utility function exists, and we know what it is, then it means that we can simply plug it into
the model and precisely replicate the observed choices of the consumer. To put it another way,
the theory and the data are consistent if there exists a well-behaved utility function which can
provide perfect within-sample fit of the consumption/demand data. Note that to allow for
easy comparison this definition simplifies with β = 1 to the rationalisability definition given
6
by Browning (1989) for the exponential discounting model.
We can now give the following result.
Proposition 1 The following statements are equivalent.
(1) The data {rt , pt , ct }t∈τ satisfy the self-aware quasi-hyperbolic discounting model.
(2) There exist constants δ ∈ [0, 1] {ψt ≥ 0}t∈τ such that
0≤
X ψt
ρ0 (cs − ct )
δt t
∀σ ⊆ τ
(RS1)
∀s,t ∈σ
1 ≤ ψt−1 ≤ ψt
where ρkt = pkt /
Qt
i=1 (1
∀t
(RS2)
+ ri ) .
The interpretation of ψt in the proposition is ψt =
Qt
i=1 (1/ [1
− (1 − β) µi ]). Proposition
1 is an equivalence result. It says that if one can find a suitable path of increasing constants
and a δ discount rate such that restrictions (RS1) and (RS2) hold, then the data are consistent
with the theory and there does indeed exist a well-behaved utility function which gives perfect
within-sample rationalisation of the data. Conversely if such constants and a δ discount rate
do not exist then there is no theory-consistent utility representation. Restriction (RS1) is a
cyclical monotonicity condition which is an implication of the concavity of the instantaneous
utility function16 and the Euler equation on the equilibrium path for this consumer. Restriction (RS2) is related to the way in which the marginal utility of consumption evolves over
time for the hyperbolic consumer.
The empirical problem is thus a question of determining whether a set of increasing constants and a δ discount rate which satisfy (RS1) and (RS2) can be found. These restrictions
are computationally quite straightforward. The important feature to note is that the restrictions are linear conditional on δ. This means that, for any choice of δ discount rate, the
existence or non-existence of the ψt terms can be efficiently determined in a finite number
of steps using phase one of a (simplex method) linear programme. The issue is then simply
one of conducting an arbitrarily fine grid search for the δ discount rate and running a linear
programming problem to check the conditions at each node.
It is useful to show how Proposition 1 relates to some important existing results so we give
two corollaries of Proposition 1 below.
Corollary 1 If condition (RS1) holds, then the data also satisfy the Generalised Axiom of
Revealed Preference (Afriat (1967), Diewert (1973), Varian (1982)).
Corollary 2 If ψt = ψt+1 in condition (RS1) then the data satisfy the conditions for the
exponential model described by Browning (1989).
Corollary 1 says that the hyperbolic discounter satisfies GARP. That is to say, regardless
of how the matter of the allocation of a budget to each period is handled, he acts as if he is
16
Rockafellar, (1970, Theorem 24.8)
7
maximising a stable well-behaved instantaneous utility function within each period. This is
essentially due to the stability of within-period tastes and intertemporal separability of the
hyperbolic consumer’s preferences which is necessary and sufficient for the second stage of
two-stage budgeting (Gorman (1959)).
The second corollary concerns the relationship with the exponential model. It shows that
the conditions in Browning (1989) for the exponential model can be regarded as a special case
of the conditions for the current model. This result is useful computationally if one wants to
compare the predictive performance of the models (see below).
Another useful aspect of the connection with the results in Browning (1989) is that it is
relatively simple to see how to extend Proposition 1 to the case where there is uncertainty.
Under certainty, (RS1) uses the fact that marginal utility of wealth, λt , evolves according to
λt =
t
1
λt−1
ψt Y
1
=λ t
δ (1 + rt ) [1 − (1 − β) µt ]
δ
(1 + ri )
i=1
and so λ can be cancelled out. As is well known, under uncertainty the consumer is modelled
as adopting expected utility as the objective function conditional on the information available
at time t. The key empirical implication of this version of the model in the exponential discounting case is that, in contrast to the perfect foresight case where it is flat, the marginal
utility of lifetime income follows a martingale E[λt |λt−1 , ...., λ1 , λ] =
1
δ(1+rt ) λt−1
(Hall (1978)).
With exponential discounting ψt = 1 ∀t but its presence in (RS1) means it will pick up the
stochastic part of marginal utility, and hence the stochastic version of Corollary 2 has ψt following a martingale: E[ψt |ψt−1 , ...., ψ1 ] = ψt−1 . Thus under uncertainty the test of exponential
discounting in Browning (1989) is simply the condition (RS1) plus E[ψt |ψt−1 , ...., ψ1 ] = ψt−1 .
Likewise to allow for uncertainty in Proposition 1 (i.e. with hyperbolic discounting) we use
1
λt−1
E[(1 − (1 − β) µt ) λt |λt−1 , ...., λ1 , λ] =
δ (1 + rt )
1
1
λt−1 + (1 − β) Cov (µt , λt )
=⇒E[λt ] =
E[(1 − (1 − β) µt )] δ (1 + rt )
(removing the conditional notation for simplicity). Since E[(1 − (1 − β) µt )] ≤ 1 this implies
E[λt |λt−1 , ...., λ1 , λ] ≥
1
λt−1
δ (1 + rt )
as long as (1 − β) Cov (µt , λt ) is not large and negative. There is no reason to expect a
relationship between the marginal utility of wealth and how a shock to wealth is allocated
across remaining periods: for example, in the standard esponential discounting model (with
discount rate equal to interest rate for simplicity) a transitory shock to wealth is divided
equally among remaining periods of life and so µt and λt are independent. Thus to allow
for uncertainty for the hyperbolic discounting case we just require that (RS2) is replaced by
the condition that ψt follows a sub-martingale (again ψt picks up the stochastic element of
8
marginal utility):
E[ψt |ψt−1 , ...., ψ1 ] ≥ ψt−1
∀t.
(RS2’)
A further interesting result is that it does not appear to be possible separately to learn much
about the the additional discount factor which is the distinguishing feature of the hyperbolic
consumption model in the sense that if the data satisfies the model then all that it is possible
to establish about β is that it is less than 1.
Corollary 3 If the data satisfies conditions (RS1) and (RS2) then it is consistent with any
β ∈ (0, 1).
This result would seem to indicate that, nonparametrically at least and with consumption
data of the type we focus on, whilst it is possible to establish that a consumer has quasihyperbolic preferences, it is not in general possible to say anything more about their β discount
factor. This result does not mean that parametric assumptions cannot provide means of
delivering β but it does show that those parametric assumptions are likely to be crucial.
2.2
The naive individual
In this section we consider the case of an individual who is a hyperbolic discounter but wrongly
assumes that his future self will simply fall into line with the consumption plan which he
maps out. The literature (e.g. O’Donoghue and Rabin (1999)) describes this as naivety and
means by this that the current self knows themselves to be a hyperbolic discounter with an
inclination for immediate gratification, but believes that future selves do not have presentbiased preferences but will behave as exponential discounters with β = 1. That is, in period
t they maximise
u (ct ) + β
T −t
X
δ i u (ct+i )
i=1
but believe that in periods τ = t + 1...T − 1 their future selves will maximise
u (cτ ) +
T
−τ
X
δ i u (cτ +i )
i=1
The implications for the first order condition and Euler equations of individuals who behave
in this way are given in the next Lemma. This shows that the Euler equation, for example,
for this type of individual is structurally very similar to that of an exponential discounter; the
difference is that the combination of the two discount rates produces an effective discount rate
which is higher than the standard case with the implication that such an individual will run
down their assets faster than the pure exponential discounter to the extent that β < 1.
Lemma 2 On the equilibrium path:
∂u
= (βδ)−t λρkt
∂ckt
∀k
9
and the corresponding Euler equation is
∂u
pk ∂u
= βδ (1 + rt+1 ) kt
k
∂ct
pt+1 ∂ckt+1
where ρkt = pkt /
Qt
i=1 (1
∀k
+ ri ) denotes the discounted price of the kth good, λ is a strictly
positive constant,β and δ lie in the interval [0, 1].
We are interested in the conditions under which observations on the interest rate, spot
prices and consumption decisions may be rationalisable by the naive version of the hyperbolic
model. Paralleling Definition 1 we describe the requirements for rationalisability below. Once
more this reduces to the case of exponential discounting considered in Browning (1989) when
β = 1.
Definition 2 The quasi-hyperbolic discounting model with naive beliefs rationalises the data
{rt , pt , ct }t∈τ if there exists a locally non-satiated, differentiable and concave instantaneous
utility (felicity) function u (.) and constants λ > 0 , β ∈ [0, 1], δ ∈ [0, 1] such that
∂u
= (βδ)−t λρkt
∂ckt
where ρkt = pkt /
Qt
i=1 (1
∀k
+ ri ).
The following proposition establishes the necessary and sufficient conditions for the naive
case.
Proposition 2 The following statements are equivalent.
(1) The data {rt , pt , ct }t∈τ satisfy the naive quasi-hyperbolic discounting model.
(2) There exist constants δ ∈ [0, 1] ,β ∈ [0, 1] such that
0≤
X 1
ρ0 (cs − ct )
dt t
∀σ ⊆ τ
(RN1)
∀s,t ∈σ
where ρkt = pkt /
Qt
i=1 (1
+ ri ) and d = βδ.
Once more the Proposition is an equivalence result: if there exists a constant d such that
the inequality in (RN 1) holds then the data are precisely replicable by the naive hyperbolic
model; if such a constant does not exist then the data, as they stand are not rationalisable.
The computational problem is again straightforward: conditional on a choice of d the condition
is simply a matter of checking a finite number of inequalities or an equivalent linear program.
Both Lemma 2 and Definition 2 are, as noted above, very similar to the corresponding
results for the exponential individual. In particular, in contrast to the self aware hyperbolic
consumer where the effective discount rate is endogenous, here it is not and the effective
discount rate is just βδ. This delivers the following corollary.
10
Corollary 4 The following statements are equivalent.
(1) The data {rt , pt , ct }t∈τ satisfy the naive quasi-hyperbolic discounting model.
(2) The data {rt , pt , ct }t∈τ satisfy the exponential discounting model.
The corollary shows that the nonparametric empirical implications of naive hyperbolic
behaviour for the data {rt , pt , ct }t∈τ are identical to those of the exponential model: if the
data are consistent with exponential behaviour they are consistent with naive hyperbolic
behaviour too and it is not possible to distinguish β and δ in (RN1) with standard observational
consumption data. In fact the only difference between exponential discounters and naive
hyperbolic discounters is one of degree rather than of kind: the naive hyperbolic consumer
behaves exactly like an exponential consumer with a high discount rate (βδ < δ). It might
be a little surprising therefore that an unsophisticated hyperbolic individual who fails to
recognise the inconsistency in their own preferences will end up behaving in a way which,
to the researcher, looks perfectly time-consistent Nonetheless the intuition is fairly clear:
the naive consumer lays a series of consumption plans but can only enact the first period’s
consumption decision; they therefore effectively make a series of one-period decisions. The
extension to allow for uncertainty simply follows Browning (1989) and replaces 1/(βδ)t in
(RN1) with λt /(βδ)t where λt follows a martingale process.
2.3
Summary
To summarise: we have presented revealed preference conditions for the self-aware and the
naive versions of the quasi-hyperbolic consumption model. These are necessary and sufficient
conditions for the agreement between the theory and data recording interest rates, spot prices
and consumption decisions. With further assumptions on the precise form of the model one
could derive further restrictions, but the ones presented in Propositions 1 and 2 and the
corollaries presented exhaust the nonparametric content of the theory, i.e. those implications
which do not follow from specific functional form assumptions.
The next section investigates the ideas discussed above using a household level consumption
panel dataset and looks at the performance of the hyperbolic model compared to an atemporal
model in which the budget in each period is parametric and the exponential discounting model
(using the characterisation in Browning (1989)).
3
Empirical Illustration
The data used here to investigate the empirical implementation of the ideas outlined above is
the Spanish Continuous Family Expenditure Survey (the Encuesta Continua de Presupuestos
Familiares - ECPF). The ECPF is a quarterly budget survey of Spanish households which
interviews about 3,200 households every quarter. These households are randomly rotated at
a rate of 12.5% each quarter. Thus it is possible to follow a participating household for up to
eight consecutive quarters. This dataset is a much studied survey which has often been used
11
for the analysis of intertemporal models and particularly, latterly, the analysis of habits models
(for example, Browning and Collado (2001, 2004)). The data used here are drawn from the
years 1985 to 1997 and are the selected sub-sample of couples with and without children, in
which the husband is in full-time employment in a non-agricultural activity and the wife is out
of the labour force (this is to minimised the effects of nonseparabilities between consumption
demands and leisure which the empirical application does not otherwise allow for). The
dataset consists of 21866 observations on 3134 households. The data record household nondurable expenditures and these are disaggregated into six commodity groups: food consumed
at home, alcohol, tobacco, clothing and footwear, restaurants and bars and domestic energy.
The discounted price data are calculated from published prices aggregated to correspond to
the expenditure categories and the average interest rate on consumer loans. It should also be
noted that these results treat the household as a unitary entity and abstracts from issues to
do with collective household behaviour.17
3.1
Consistency Results
We examined the consistency of the data with each of there models: atemporal utility maximisation (i.e. a GARP test), the exponential consumption model and the sophisticated quasihyperbolic consumption model. We examine the data for consistency for each of these models
using the data for each individual household, one at a time18 . It is important to emphasise
that the data are never pooled across households so we therefore allow for complete preference heterogeneity within the classes of models studied: households may differ with respect to
whether they are rationalisable by a given model, their discount rates and their instantaneous
utilities. Since we only have a handful of observations on each household a nonparametric
statistical test of the martingale/sub-martingale hypothesis (for example along the lines of
Phillips and Jin (2013)) would have no power to reject. We therefore focus on the perfect
certainty version of the models in our empirical work. Table 1 reports the pass rates for each
model considered.
Table 1: Rationalisability results
Model
Atemporal
Exponential
Quasi-hyperbolic
discounting
discounting
Pass Rate
0.9569
0.0045
0.5281
(Std. Error)
(0.0036)
(0.0012)
(0.0089)
From Table 1 we see that the atemporal model19 rationalises the behaviour of 95% of the
sample. The exponential discounting/naive model by contrast rationalises very few households
17
For a discussion of the issues raised by collective models of households in a revealed-preference framework
see Cherchye et al (2007) and especially Adams et al (2012) who consider the coexistence of exponential
discounters with differing discount rates within a household as another potential source of time-inconsistency
in aggregate (household) behaviour.
18
For the intertemporal model we conducted a grid search for δ over the unit interval with a grid spacing of
0.001 - this means that formally the pass rates for these models are lower bounds.
19
maxq u (q) subject to p0t q = p0t qt
12
(less than half of one percent). Given that within-period consistency with utility maximisation
(GARP) is a necessary but not sufficient condition for the exponential model (see Browning
(1989)) this disparity must therefore be due, in the main, to the intertemporal behaviour
displayed by the households in the sample. The hyperbolic model falls between these extremes
- just over half of the sample display behaviour which is precisely rationalisable by hyperbolic
preferences. The standard errors for each of these statistics is given in brackets in Table 1. In
each case the effects of sampling variation appear to be quite modest and we would therefore
expect the proportions of households consistent with each model in another random sample
of a similar size from this population to be close to those in Table 1.
3.2
Model comparison
The results from the consistency tests are fairly stark and are statistically significant, but are
they economically significant in the sense of providing useful information about the various
models considered? Can we conclude that the atemporal model is the best model followed by
the quasi-hyperbolic model and that the exponential model stands condemned?
It is important to remember when looking at the pass rates that these models are not
equally demanding. The atemporal model of utility maximisation does not consider intertemporal planning at all and takes the budget allocated to each period as parametric. It only
requires that within-period preferences are rational and stable. It has no restrictions whatsoever concerning how spending is allocated across time and therefore, as long as within-period
preference satisfy GARP, any intertemporal allocation is, in that sense, rationalisable with
the atemporal model. The exponential model is much more demanding. It constrains both
within-period and intertemporal choices: within-period choices must be rational and stable,
and intertemporal choices must be time-consistent. The hyperbolic model is, in a sense, intermediate. It too requires within-period rationality and stability (as shown in Corollary 1),
but whilst it does not require time-consistency, intertemporal behaviour is constrained by the
form of the quasi-hyperbolic Euler equation. To view the comparison of the exponential with
the quasi-hyperbolic model another way; compared to the exponential model, quasi-hyperbolic
preferences introduce a free parameter - the β discount rate. Thus, whilst much is sometimes
made of the ability of the hyperbolic model to explain observed behaviour which the exponential model cannot20 , this should come as little surprise. How, then, should we make sense
of the results in Table 1?
In their revealed preference guise, stripped of special functional form assumptions, all
three models generate restrictions in the form of sets of choices which are consistent with the
model of interest (for example, given any collection of budget constraints there will be a set of
demands which satisfy GARP). To investigate the performance of models which predict sets,
it is useful to think about two objects. The first is the feasible outcome space (for example, the
set of choices which satisfy the intertemporal budget constraint) which we will denote F . The
second is the set of model-consistent choices (for example, that set of feasible choices which
20
See Frederick et al (2001) for example.
13
satisfy the restrictions of the model), denoted P . Clearly the model-consistent choices are a
subset of the feasible choices: P ⊆ F . When one conducts a particular empirical revealed
preference test one is, in essence, checking to see whether the observed choices lie within P .
With this in mind it becomes clear that it is necessary to allow for the relative size of
the predicted/theoretically consistent subset. The essential idea - which is due to Selten and
Krischker (1983) and Selten (1991) - is that if the set of observations consistent with the
model(P ) is very large relative to the set of behaviours which the consumer could possibly
display (F ) then we should be little surprised if we find that many of the observed choices lie in
P - they could hardly have done otherwise. For example, if we are testing the atemporal model
and the collection of budget constraints never cross then all feasible choices, necessarily, satisfy
GARP: it would be impossible to make an observation which wasn’t in P because P = F .
This means that empirical fit alone (the proportion of the sample which passes the relevant
test) is not a sufficient basis for ranking the performance of alternative theories: if it were
then no theory could out-perform a meaningless theory which can rationalise any behaviour.
A better approach would be to consider the trade-off between the pass rate and some sort
of measure of how demanding the theory is. Following Selten (1991) let a denote the size
of the theory-consistent subset P , relative to the outcome space F for the model of interest.
The relative area of the empty set is zero and the relative area of all outcomes is one so
a ∈ [0, 1]. Now suppose that we have some choice/outcome data. Let r denote the pass rate;
this is simply the proportion of the data that lies in P and hence satisfies the restrictions of
the model of interest (i.e. the numbers in Table 1). Selten (1991) argues that demanding
theories are characterised by small values for a; and empirically successful theories combine
small values of a with a high degree of agreement between the data and theory (large r). He
also argues that the trade-off between the ability to fit the data and the restrictiveness of the
theory should be the difference measure21 : r − a. In what follows we take a somewhat different
approach.
Since both the relative size of the theory-consistent set and the empirical pass rate satisfy
all of the necessary properties of probabilities22 we can be justified in thinking about the problem of comparing them as the problem of comparing probability distributions. Here {a, 1 − a}
is a bivariate probability distribution describing the probability that random uniform choice
over the feasible set will satisfy the restrictions of the model. Similarly {r, 1 − r} is a bivariate probability distribution describing the probability that a subject drawn at random with
uniform probability from the population will exhibit behaviour consistent with the model. A
simple way in which to make a comparison between these two distributions in units which are
meaningful is to use the Kullback-Leibler divergence
KL (r, a) =
X
i
21
ri log2
ri
ai
See also Beatty and Crawford (2010) for an application of this to revealed preference test.
They are non-negative, sum to one and the area of two none-overlapping subsets is the sum of their
individual areas.
22
14
where i indexes possible outcomes. The Kullback-Leibler divergence measures the information
(in bits) conveyed by the empirical results relative to what one would expect from the theoretical restrictions - in a well-defined formal sense it measures how “surprising” the outcome
of the test is given the restrictiveness (or otherwise) of the theory of interest23 . For example,
suppose that the pass rate was 0.7 and this matched the size of theory-consistent set predicted
by the model precisely (i.e. the proportion of all possible observed choices which satisfy the
theory was also 0.7) then the observed pass rate would convey no surprise at all and the data
would generate zero bits of information about the empirical performance of the model. If,
on the other hand, the set of outcomes which are rationalisable represent very little of the
outcome space (a → 0) (i.e. the model makes very precise predictions) and most of the data
generally satisfy the theoretical restrictions (r → 1) or even if the reverse were true (r → 0)
then the outcome of the empirical test is extremely informative and KL (r, a) is large: in
the former case it is informative of the spectacular success of the theory, in the latter it is
informative about equally spectacular failure.
Table 2: Kullback-Leibler Divergence
Model
KL (r, a)
Atemporal
Exponential
Quasi-hyperbolic
discounting/naive
discounting
0.940
10.342
240.691
(43.876)
(5.014)
(26.639)
Table 2 presents the Kullback-Leibler results for the three models. The numbers represent
the number of bits of information conveyed by the results. To give a sense of scale it might
be helpful to bear in mind that one flip of a coin believed to be fair contains one bit of
information, 3134 flips of a coin which was believed to be fair and which all came up heads
would (i) be very surprising and (ii) would contain 0.39 kilobytes of information about the
true probability of a head, and any number of flips of a coin which contained equal numbers
of heads and tails would contain zero bits of information. The very high pass rate of the
atemporal model is revealed to be rather like this last example and quite uninformative about
the success or otherwise of that model - and indeed the standard error indicates that the nonzero value is very likely to be due to sampling variation: another sample could easily produce
zero information about the model. The KL divergence being essentially zero indicates that
the predictive success of the model is due, almost entirely, to its permissiveness. Turning to
the comparison of interest, that between the exponential and the hyperbolic model, we see
that even allowing for the less restrictive nature of the hyperbolic model compared to the
exponential model, the performance of the hyperbolic model is far better than the exponential
model. Furthermore this difference between the performances of the models which we observe
in this sample of households appears to be unlikely to have arisen simply by chance through
random sampling. The better performance of the quasi-hyperbolic model does not appear to
be a simple mechanical artifact associated with a free parameter.
23
The surprisal of an event with probability pi is proportional to log2 (1/pi ) so the KL divergence is interpretable as the average surprisal of the empirical performance of the model compared to the performance of a
uniform random number generator acting over the set of feasible outcomes.
15
4
Conclusion
We provide the a revealed preference characterisation of quasi-hyperbolic consumption behaviour by a self-aware consumer who recognises the tension between their short-term and
long-term selves and a naive consumer who does not. This class of models may allow for
time-inconsistent behaviour but we show that, despite this, there is enough consistency in
their behaviour that the model is meaningful in the Samuelsonian sense - it is refutable on the
basis of observables alone. Furthermore the empirical conditions are quite easy to apply since
they require nothing more complicated than linear programming. In an empirical application
we consider in some detail how to interpret the revealed preference performance of alternative
models which differ in their restrictiveness. We suggest the Kullback-Leibler divergence as a
means of doing this. This seems to be quite a fruitful way of describing the performance of the
models in question - and possibly economic models in general. We find that, for these data,
the hyperbolic model provides the best description of behaviour of the models considered here.
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18
Appendix - Proofs
Proof of Lemma 1 and Lemma 2
Let β denote the additional discount factor associated with hyperbolic discounting. Denote
savings as St and income as Yt . In period t, person t maximises:
V (At ) = max u (ct ) + β
ct ...cT
s.t.
T −t
X
δ i u (ct+i )
(1)
i=1
p0j cj + Sj = Yj + Aj
Aj = (1 + rj ) Sj−1
j = t, ..., T
but believes that persons τ = t+1...T (his future selves) will use a different quasi hyperbolic
b and thus will maximise
discounting factor, β,
W (Aτ ) = max u (cτ ) + βb
cτ ...cT
s.t.
T
−τ
X
δ i u (cτ +i )
(2)
i=1
p0j cj + Sj = Yj + Aj
Aj = (1 + rj ) Sj−1
j = t + 1, ..., T
We can re-write the value function in equation (1) as
V (At ) = max u (ct ) + δu (ct+1 ) + β
ct ...cT
T −t
X
At+1
(3)
i=2
s.t.
δ i u (ct+i ) + βδu (ct+1 ) − δu (ct+1 )
β
W (At+1 ) − β
= max u (ct ) + δ
ct
βb
= (1 + rt ) Yt + At − p0t ct
1
− 1 u (ct+1 )
βb
and that in equation (2) as
W (Aτ ) = max u (cτ ) + δu (cτ +1 ) + βb
cτ ...cT
s.t.
Aτ +1
T
−τ
X
b (cτ +1 ) − δu (cτ +1 )
δ i u (cτ +i ) + βδu
(4)
i=2
h
i
= max u (cτ ) + δ W (Aτ +1 ) − 1 − βb u (cτ +1 )
cτ
= (1 + rτ ) Yτ + Aτ − p0τ cτ
The first order condition from equation (3) is
"
#
X
K
k
∂c
∂ut
β
1
∂u
t+1
− δ (1 + rt+1 ) pkt
WAt+1 − β
−1
=0
∂ckt
∂ckt+1 ∂At+1
βb
βb
k=1
19
∀k
(5)
and the envelope theorem gives
"
V At
β
= δ (1 + rt+1 )
WAt+1 − β
βb
#
X
K
1
∂u ∂ckt+1
−1
∂ckt+1 ∂At+1
βb
k=1
∀k
(6)
Equations (5) and (6) give
pkt VAt =
∂ut
∂ckt
∀k
(7)
From beliefs about the future (equation (4)) we get first order conditions that person t believes
will be used at t + 1
#
"
K
X
k
∂c
∂u
∂u
t+2
=0
− δ (1 + rt+2 ) pkt+1 WAt+2 − 1 − βb
k
∂A
∂ckt+1
∂c
t+2
t+2
k=1
∀k
(8)
and the envelope theorem gives
"
WAt+1 = δ (1 + rt+2 ) WAt+2
K
X
∂u ∂ckt+2
− 1 − βb
∂ckt+2 ∂At+2
k=1
#
(9)
Equations (8) and (9) give that person t believes that
pkt+1 WAt+1 =
∂u
∂ckt+1
∀k
(10)
Substituting equation (10) back into equation (5) gives:
∂u
pkt
=
δ
(1
+
r
)
t+1
∂ckt
pkt+1
"
β
−β
βb
#
X
K
∂ckt+1
1
∂u
k
pt+1
−1
k
b
∂A
∂c
t+1
β
t+1
∀k
(11)
k=1
For the sophisticated agent, βb = β and hence equation (11) reduces to the Euler equation
in Lemma 1.
Define
µt+1 =
K
X
pkt+1
k=1
∂ckt+1
∂At+1
Note that this is the period t + 1 marginal propensity to spend out of wealth. Given
consumption in each period is normal then it follows that
µt ∈ [0, 1]
∀t
Solving for ∂u/∂ckt+1 gives
−1
pkt+1 β
∂u
∂u
1
1
=
−
β
−
1
µ
t+1
∂ckt+1
∂ckt δ (1 + rt+1 ) pkt
βb
βb
20
∀k
and solving recursively gives
t
t
Y
∂u
1
1
∂u pkt 1 Y
h
i
=
t
k
k
k
∂ct
∂c0 p0 δ i=1 (1 + ri ) i=1 β − β 1 − 1 µi
b
b
β
∀k
β
Using (7) dated in t = 0 and denoting VA0 = λ
λ=
∂u 1
∂ck0 pk0
∀k
On substitution this gives the condition
t
∂u
1 kY
1
i
h
= λ t ρt
k
β
1
δ
∂ct
i=1 b − β b − 1 µi
β
∀k
(12)
β
For the sophisticated agent, βb = β and hence equation (12) reduces to
t
∂u
1 kY
1
= λ t ρt
k
δ
[1 − (1 − β) µi ]
∂ct
i=1
∀k
which is Definition 1. Reinserting the expression for µi gives the first equation of Lemma 1.
For the naive agent, βb = 1 and hence equation (12) reduces to
∂u
= λ (βδ)−t ρkt
∂ckt
∀k
which is the first equation of Lemma 2.
Proof of Proposition 1
(T ) ⇒ (R)
Definition 1 and Lemma 1 state that
t
∂u
1 kY
1
= λ t ρt
k
δ
[1
−
(1
− β) µi ]
∂ct
i=1
Denote
ψt =
t
Y
i=1
1
[1 − (1 − β) µi ]
Because β ∈ [0, 1] and µt ∈ [0, 1] this imples that
1
≥1
[1 − (1 − β) µi ]
Therefore
ψt ≥ ψt−1 ≥ 1
21
∀k
which is condition (RS2).
Rewriting the first order condition on the equilibrium path in vector notation gives
∇u (ct ) = λ
1
ρt ψt
δt
Concavity of the instantaneous utility function gives
u (cs ) ≤ u (ct ) + ∇u (ct )0 (cs − ct )
∀s, t ∈ τ
Substituting in the first order conditions means there exist real numbers {ut }t∈τ such that
us ≤ ut + λ
ψt 0
ρ (cs − ct )
δt t
Sum over any subset of observations σ ⊆ {1, ..., T } gives
0≤λ
X ψt
ρ0 (cs − ct )
δt t
∀σ ⊆ τ
∀s,t ∈σ
Since λ > 0 this implies
0≤
X ψt
ρ0 (cs − ct )
δt t
∀σ ⊆ τ
∀s,t ∈σ
which is condition (RS1)
(R) ⇒ (T )
Restriction (RS1) is a cyclical monotonicity
n
o condition (Rockafellar, 1970, Theorem 24.8).
ψt
Cyclical monotonicity for the data δt ρt , ct
implies that there exists a concave, strictly
increasing (utility) function u (c) such that
t∈τ
∂u
ρkt
=
ψt
δt
∂ckt
∀k, t
(the non-differentiable case is similar but involves super-gradients). Setting λ = 1 we also
have
∂u
ρkt
=
λ
ψt
δt
∂ckt
∀k, t
for all t ∈ τ . We therefore have a suitable utility function and the parameters δ and λ. We
now need to show that we can take the sequence
1 ≤ ψt ≤ ψt+1
and find β ∈ [0, 1] and a sequence of µt ∈ [0, 1] such that
ψt =
t
Y
i=1
1
[1 − (1 − β) µi ]
22
Given the sequence of ψt terms we can take the ratio
ψt−1
= 1 − (1 − β) µt
ψt
ψt−1
1−
= (1 − β) µt
ψt
and for t = 1 we have
ψ1 =
1−
1
[1 − (1 − β) µ1 ]
1
= (1 − β) µ1
ψ1
Since µt ∈ [0, 1]this implies that we must have
0<β<
ψt−1
ψt
Since ψt−1 ≤ ψt we know that this implies
(1 − β) µt ∈ [0, 1]
This provides T − 1 equations with T unknowns so we can always find a suitable β and µt for
t = 2, ..., T. For t = 1 we have
ψ1 =
1
[1 − (1 − β) µ1 ]
which we can solve for µ1 by substitution. There therefore exists a concave, strictly increasing
(utility) function u (c) and positive constants λ, β,{µt }t=1,...,T such that
t
1 kY
1
∂u
= λ t ρt
k
δ
[1 − (1 − β) µi ]
∂ct
i=1
∀k, t
which is the definition of rationalise in in Definition 1
Proof of Corollary 1
Restriction (R1) is a cyclical monotonicity conditionn(Rockafellar,
1970, Theorem 24.8). By
o
ψt
implies that there exists a
Afriat’s Theorem cyclical monotonicity for the data δt ρt , ct
t∈τ
n
o
concave, strictly increasing (utility) function u (c) such that the data ψδtt ρt , ct
satisfies
t∈τ
GARP. Since
ψt 0
ψt
ρt ct ≥ t ρ0t ct ⇒ p0t ct ≥ p0t ct
t
δ
δ
n
o
satisfies GARP so do the data {pt , ct }t∈τ .
it follows that the data ψδtt ρt , ct
t∈τ
23
Proof of Corollary 2
Immediate by comparison with Browning (Proposition 1, 1989) (augmented to allow for time
discounting which Browning does not explicitly consider).
Proof of Corollary 3
From the proof of sufficiency in Proposition we show that we can take the sequence of constants
1 ≤ ψt ≤ ψt+1 and derive a set of T − 1 equations with T unknowns such that
1−
ψt−1
= (1 − β) µt
ψt
Thus β is under-determined.
Proof of Proposition 2
(T ) ⇒ (R)
Definition 2 and Lemma 2 state that
∂u
1 k
=λ
ρt
k
∂ct
(βδ)t
∀k
Define d = βδ. Rewriting this first order condition in vector notation gives
∇u (ct ) = λ
1
ρt
dt
Concavity of the instantaneous utility function
u (cs ) ≤ u (ct ) + ∇u (ct )0 (cs − ct )
∀s, t ∈ τ
Substituting in the first order conditions means there exist real numbers {ut }t∈τ such that
us ≤ ut + λ
1 0
ρ (cs − ct )
dt t
Sum over any subset of observations σ ⊆ {1, ..., T } gives
0≤λ
X 1
ρ0 (cs − ct )
dt t
∀σ ⊆ τ
∀s,t ∈σ
Since λ > 0 this implies
0≤
X 1
ρ0 (cs − ct )
dt t
∀σ ⊆ τ
∀s,t ∈σ
which is condition (RN 1)
(R) ⇒ (T )
Restriction (RN 1) is a cyclical monotonicity condition (Rockafellar, 1970, Theorem 24.8).
24
Cyclical monotonicity for the data
1
dt ρt , ct t∈τ
implies that there exists a concave, strictly
increasing (utility) function u (c) such that
∂u
ρkt
=
dt
∂ckt
∀k, t
(the non-differentiable case is similar but involves super-gradients). Setting λ = 1 we also
have
∂u
ρkt
=
λ
dt
∂ckt
∀k, t
for all t ∈ τ . We therefore have a suitable utility function and the parameters d and λ. Given
any d we can find two real numbers β and δ such that
βδ = d
There therefore exists a concave, strictly increasing (utility) function u (c) and positive constants λ, β and δ such that
∂u
1 k
ρt
=λ
∂ckt
(βδ)t
∀k, t
which is the definition of rationalise in in Definition 2.
Proof of Corollary 3.
Setting d = βδ this is Immediate by comparison with Browning (Proposition 1, 1989) (augmented to allow for time discounting which Browning does not explicitly consider).
25